Notes on the TAN projection

Equations 25 and 26 of Representations of Celestial Coordinates in FITS (Calabretta & Greisen, draft) describe a generalized polynomial plate model containing terms in x, y, and r up to 7th-order.

It is stated in the paper that this polynomial is sufficient to account for the complex distortions found in some optical systems.

The purpose of this note is to demonstrate this for some real-world cases.

1. BENDXY

(Click on each diagram for a more detailed PostScript image.)

The first case is that of an empirical fit made by Pat Wallace to model the distortions in the corners and towards the perimeter of UKST Schmidt plates.

The distortions were originally modelled via a combination of radial powers and trigonometric functions with three coefficients defined in a FORTRAN routine called BENDXY.

Offsets computed by BENDXY on a 40 x 40 grid over the 350mm x 350mm plate area are shown above left, greatly exaggerated. The rms deviation of the offsets is 3.7 µm and the maximum deviation is 15.4 µm.

It was found that BENDXY could be satisfactorily modelled with an rms deviation of the residuals of 0.10 µm (maximum deviation 0.47 µm) using a 7th-order polynomial, though only 10 terms in each of xi and eta were required. Residuals are shown above right, where the exaggeration is now a factor of ×10 greater than for the offsets.

2a. DEIMOS (whole field)

DEIMOS is the Deep Extragalactic Imaging Multi-Object Spectrograph, an instrument being developed by the UCO/Lick Observatory at Santa Cruz for installation on the Keck II telescope.

Steve Allen has investigated the distortions expected for the DEIMOS optical system and kindly transmitted these in analytic form.

The 10-coefficient analytic expressions were encoded in a FORTRAN subroutine for least squares analysis. Offsets computed by this routine on a 40 x 40 grid over the 8192 x 8192 pixel CCD array are shown above left, greatly exaggerated. The rms deviation of the offsets is 7.3 pixel with maximum deviation 46.3 pixel.

2b. DEIMOS (imaging subregion)

Since only a portion of the 8192 x 8192 pixel CCD array will be used for imaging I also investigated the polynomial fit for the approximate area concerned. Within this region the rms deviation of the offsets is 7.0 pixel with a maximum deviation of 22.6 pixel.

The same terms of the 7th-order polynomial were required as for the full array. However, the rms deviation of the residuals fell to 0.011 pixel (maximum deviation 0.074 pixel) which corresponds to 0.2 and 1.9 milliarcsec. Residuals are shown above right, where the exaggeration is now even greater than for the whole-field diagrams - a factor of ×400 times that of the offsets.

An adequate fit was obtained using only a 4th-order polynomial with 11 terms in xi and 6 in eta. For this the rms deviation of the residuals was 0.11 pixel with maximum deviation 0.62 pixel.

A 3rd-order polynomial with 8 terms in xi and 4 in eta produced an rms deviation of the residuals of 0.24 pixel and maximum deviation 1.5 pixel.

             BENDXY    DEIMOS
   Term     xi  eta   xi  eta
  -------   -------   -------
  1                   *
  x         *         *
  y              *         *
  r                   *
  xx                  *
  xy                       *
  yy                  *
  xxx       *         *
  xxy            *         *
  xyy       *         *
  yyy            *         *
  rrr                 *
  xxxx                *
  xxxy                     *
  xxyy                *
  xyyy                     *
  yyyy                *
  xxxxx     *         *
  xxxxy          *         *
  xxxyy     *         *
  xxyyy          *         *
  xyyyy     *         *
  yyyyy          *         *
  rrrrr               *
  xxxxxx              *
  xxxxxy                   *
  xxxxyy              *
  xxxyyy                   *
  xxyyyy              *
  xyyyyy                   *
  yyyyyy              *
  xxxxxxx   *         *
  xxxxxxy        *         *
  xxxxxyy   *         *
  xxxxyyy        *         *
  xxxyyyy   *         *
  xxyyyyy        *         *
  xyyyyyy   *         *
  yyyyyyy        *         *
  rrrrrrr             *

It can be seen that each term was used at least once in either xi or eta.

In conclusion, the 7th-order polynomial plate model appears to be more than adequate for the examples so far studied.